Are There Numbers in our Nature?

Radio Lab (WNYC) is a great podcast that has many interesting episodes on tons of different topics, from laughter to lying, from parasites to placebos. A recent episode explores numbers. Studies with babies and toddlers show that we come into this world hard-wired with instincts about numbers and math.

http://www.wnyc.org/shows/radiolab/


Stanislaus Dehaene showed very young (2-3 m.o.) babies pictures of things such as ducks or trucks. The babies were outfitted with sponges with teeny-tiny electrodes, so brain activity could be monitored as they looked at pictures with either interest or boredom. Dehaene found that even very young babies have an internal sense of number, an intuition about quantity that shows up in startled interest when the number of ducks or trucks suddenly changes dramatically.

Strangely, Dehaene's work indicates that babies' internal sense of number is logarithmic rather than integral, more about ratio than counting. In other words, to a baby, the difference between 1 and 2 is enormous compared to the difference between 8 and 9. This is because 2 is twice as large as 1, but 9 is only a small fraction larger than 8. I haven't seen all the details of the research backing up Dehaene's assertion, but I hope to track down his book, The Number Sense: How the Mind Creates Mathematics, to find out more.

A long time ago, I read Young Children Reinvent Arithmetic, by Constance Kazuko Kamii. This fascinating book describes experiments with youngsters, but without brain-imaging technology, Kamii and others naturally had to work with children who can talk rather than babies. One important message from the book is that it is a bit ham-handed of adults to insist that children work with paper-and-pencil math processes before the children have a conceptual idea of that process. Even if kids can manage to remember a particular operation or procedure, if they don't understand what they are doing, they are likely to mess it up, forget it, or at the very least decide that school arithmetic is unrelated to real life.

An example, loosely taken from Kamii's book (the details of the interaction, such as the actual numbers used, may be different):

A girl who had “learned” to carry when adding explained to a researcher how to do it.
She wrote “18 + 5” on the board in vertical form. She drew a line under the problem and then explained, “Eight plus five equals 13, so you write that here”--and she wrote the number 13 below the line. “And one plus nothing equals one, so you write that here”--and she wrote the number 1 next to the 13.
The answer now looked like 113. The researcher asked, “Can you read the answer for me?”
The little girl labored over the task a bit but finally said, “One hundred and thirteen.”
“So,” the researcher went on, “18 plus 5 equals 113?”
“Yes. Because it's called 'carrying.'”
“So, if you had 18 pieces of candy, and I gave you 5 more, you would then have 113 pieces of candy?”
The little girl laughed. “No,” she explained. “It doesn't work with candy!”

Let's please, none of us, teach math in such a way that kids think that something that doesn't work for candy or apples or other countable items DOES WORK with chalk marks on a board or numbers scribbled onto a piece of paper!

In a world where most high school and college math classes--and even the SAT test, which often lies between—accept the use of calculators, in a world in which calculators are inexpensive and ubiquitous (especially as every cell phone has one!) perhaps children don't have to practice calculating so very early and so very often. Perhaps it's time to move formal arithmetic instruction back to higher grades so that young children can use and capitalize on their innate number sense without getting confused by the imposition of procedures for which they haven't yet developed a conceptual foundation.

In other words, perhaps with math as well as reading, “Better late than early.”